Adaptive neural network prescribed performance control for dual switching nonlinear time-delay system

This paper investigates the adaptive neural network prescribed performance control problem for a class of dual switching nonlinear systems with time-delay. By using the approximation of neural networks (NNs), an adaptive controller is designed to achieve tracking performance. Another research point of this paper is tracking performance constraints which can solve the performance degradation in practical systems. Therefore, an adaptive NNs output feedback tracking scheme is studied by combining the prescribed performance control (PPC) and backstepping method. With the designed controller and the switching rule, all signals of the closed-loop system are bounded, and the tracking performance satisfies the prescribed performance.

Dual switching systems are a class of hybrid systems that have both deterministic switching subsystems and stochastic switching subsystems. Its switching mechanism is more complex than the traditional switching system, and contains both deterministic and random switching signals. Since Markov stochastic processes are often used to describe its stochastic subsystems, it is also known as switching Markov jump system 1,2 . It has been gradually applied in recent years to model systems such as network control systems 3,4 , battery energy storage system 5 , fault tolerant systems 6 , etc.
The control problem for dual switching nonlinear system has received a lot of attention. Bolzern et al. 1 and Song et al. 2 studied the almost sure stability of Markov jump linear systems. Reference 7 provided sufficient conditions in terms of matrix inequalities for mean square stability and mean square stabilizability of discretetime dual switching systems. The almost sure stability for discrete dual switching system and continuous dual switching system is studied respectively in Refs. [8][9][10] .
As the switching mechanisms for practical applications are becoming more and more complex and the requirements for the system are higher, the transient performance of the system has attracted widespread attention. To achieve the prescribed performance, the studies for switched systems mainly used neural network adaptive control 11,12 , dynamic surface control 8,13 , fuzzy control [14][15][16][17] and other schemes 18 . Time-delay, as an important factor affecting system performance, is also widespread in practical systems. Therefore, research for time-delay systems is also meaningful. The research results for time-delay systems are numerous. In order to investigate the stability of switched time-delay systems, the Lyapunov-Krasovskii functions 19,20 , average dwell time (ADT) method 21 , neural network-based adaptive control 22,23 . Figure 1 shows a dual switching system model applied to the scheduling of a network control system with packet dropout. Assume that there are M plants that are controlled by a regulator. The scheduling signal γ (t) takes values in the set M and only one plant is scheduled at a time. The data transmission over the network is affected by a stochastic fault that is modeled by a Markov process σ (t) taking values in the set N = {1, 2} . Let σ (t) = 1 represent the fault-free mode when all packets are transmitted correctly, and σ (t) = 2 represent the packet dropout mode when no packets are sent. In this dual switching model 3 , the scheduling signal γ (t) needs to be designed to satisfy the stability of the data transmission in the network. www.nature.com/scientificreports/ Motivated by the above observations, the following questions have drawn our attention: (1) How to design an adaptive controller when both unmeasurable states and time-delay exist in a dual-switching nonlinear system? (2) How to achieve stabilization when both transient stability and system stability are required? Those questions will be solved in this paper. In this paper, adaptive neural network prescribed performance control for a class of dual switching nonlinear systems with time-delay is investigated. The backstepping method, multiple Lyapunov functions, and prescribed performance theory are applied to design an adaptive neural controller. Then we prove that all signals in the closed-loop system are bounded and the steady-state and transient performance are satisfied under the joint action of the designed switching rule and the controller. The main contributions of this paper are as follows.
(1) Compared with previous works 22,23 , this paper studies a class of dual-switching nonlinear systems with more complex switching mechanisms and with both time-delay and unobservable states. The system model studied is more general and challenging than the present results. The approximation capability of neural networks is used to design adaptive controllers to stabilize the system. (2) Previous studies 9,10 focused on the stability of dual-switching systems. However, these works could not guarantee the predefined transient and steady-state performance, and none of them considered the timedelay. The control scheme proposed in this paper successfully solves the PPC problem for dual switching nonlinear systems with time-delay. (3) Adaptive laws for each subsystems were designed to reduce the conservatism introduced 14,18 by the common adaptive law for all subsystems.
The rest of the paper is organized as follows. The problem formulation and the preliminaries are presented in "Problem formulation and preliminaries" section. The main results are derived in in "Controller design" and "Stability analysis" sections. An example is shown in "Numerical simulation" section. "Conclusion" section concludes this paper.

Problem formulation and preliminaries
Problem formulation. Consider the following dual switching nonlinear system with time-delay.
w h e r e x i = (x 1 , . . . , x i ) T a n d x = (x 1 , . . . , x n ) T ∈ R n a r e t h e s y s t e m s t a t e v a r i a b l e s .
. , x n (t − τ n )) ∈ R n are the time-delay of the system states. τ i stands for a stochastic delay and has the upper bounds τ m . The nonlinear terms f σ (t) and the nonlinear time-delay terms h γ (t) σ (t) are continuous and unknown. u

Assumption 1
The reference signal y d is smooth and y d , ẏ d and ÿ d are bounded. There exists a positive constant Y D such that y d ,ẏ d ,ÿ d belongs to the compact y , where y = y d ,ẏ d ,ÿ d : Assumption 2 For i = 1, . . . , n , the nonlinear function h i,σ (t) (x i (τ i )) satisfies the following inequality: (1) www.nature.com/scientificreports/ Remark 1 Assumption 1 is common in signal tracking with specified performance 23,24 . Assumption 2 is a general assumption in nonlinear time-delay systems 25,26 . In fact, for any continuous function h(x 1 , . . . , x n ) : R n → R , there exist positive smooth functions 27 .
Prescribed performance control. To achieve the prescribed transient and steady state behavioral bounds on the tracking errors s 1 (t) = y(t) − y d (t) , guaranteeing the objective is equivalent to It can be claimed that prescribed performance shown in (2) is guaranteed if the ν(t) is bounded.

Radial basis function neural network.
Let F(z) be an unknown continuous function defined on a given compact set z . The RBFNN can be used to approximate F(z) on the compact set z . It can be expressed as where ξ(z) is the approximate error, which is bounded. θ = θ 1 θ 2 · · · θ l , l > 1 denotes weight vector, and T is the basis function vector with ϕ i (z) being the Gaussian function in the form where c i is the center of the radial basic function and σ i is the width of the Gaussian function.

Controller design
To begin with the controller design procedure, let us introduce the following coordinates transformation: where e i is the error, z i is the output state that is obtained through first-order filter, α i−1 is the virtual controller. The first-order filters are given as where ς i is a positive design parameter.
Step 1 For i = 1 , ė 1 is directly obtained from (1) and (3), From Assumption 2, we can get that According to (3), x 2 = e 2 + χ 2 + α 1 . Then, it holds  (5) and (6) into (4), Using the Young's Inequality, one has By (7) and (8), we have For each stochastic subsystem, design the adaptive law and the virtual control as With the adaptive law (9) and (10), V γ k 1,σ can be recalculated as Step i: (1) and (3), we have The Lyapunov function can be chosen as follows Assume the k-th determinate subsystem is activated, denoted by γ (t) = γ k . Then the derivative of V i satisfies that By the Assumption 2 , the inequality as follows is holds: By substituting (13) and (14) into (12), we have The RBF neural network is used to fit the curve of unknown function F i,σ X i , the approximation error ξ γ k i,σ X i is bounded and satisfies ξ γ k i,σ X i ≤ ξ i . For each determinate system and stochastic system, we design the adaptive controller as Invoking (15) and (16) can be equivalently rewritten as Step n: The actual controller will be designed in the final step. By (1) and (3), the error can be computing as n,σ (t) (x(τ )) −ż n . Consider the Lyapunov function candidate as where θ n = θ n −θ n , and l n = 1 2 n j=1 e −k n( t−τ j ) t t−τ j e k n s β 2 n,j x j (s) ds . The definition of θ n is given later. Assume the k-th determinate subsystem is activated, (18) can be derived according to (17).

According to Assumption 2, it can be checked that
The control input defined as u γ k σ = α n for convenience. Following a similar process to step i, we can obtain that where F γ k n,σ X = f γ k n,σ (x) + 1 2e n n j=1 e k n τ j β 2 n,j x j (t) −ż n and the adaptive law is designed as Now, we design the virtual control and actual control input as follows, i,σ , then, with the adaptive laws, the virtual control and actual control input, we can deduce that With the transformation of the virtual controller in (3), There exists a non-negative continuous V γ k n,σ ≤ e n F γ k n,σ X + e n g γ k n,σ (x)α n + 1 2 e 2 n + χ nχn − θ γ k n,σ Tθ γ k n,σ µ γ k n,σ − k n l n ≤ e n θ γ k n,σ T ϕ γ k n,σ X + ξ γ k n,σ X − θ γ k n,σ Tθ γ k n,σ µ γ k n,σ + e n g γ k n,σ (x)α n + 1 2 e 2 n + χ nχn − k n l n , (20) θ γ k n,σ = µ γ k n,σ e n ϕ γ k n,σ X .

Stability analysis
This section is devoted to analyzing and proving the stability of the entire closed-loop system. By designing the switching strategy, the controlled daul switching system achieves stability, including signal boundedness and convergence of the tracking error, and the results are summarized in Theorem 1. Let l 0 , l 1 , . . . , l q be the stochastic switching time sequence during [t k , t k+1 . The switching timing diagram is shown in Fig. 2. Given , obviously, it's a piecewise differentiable along the solutions of system (1) on [l i , l i+1 ) . By multiplying both sides of (24) by e ct , we can get Theorem 1 Consider the nonlinear dual switching time-delay system described by (1) with Assumptions 1-3. Controllers (10,21), adaptive laws (9,16,20) are designed. Suppose that there exist class K functions α 1 , α 2 ∈ K ∞ , given constants ϑ . . , m} , such that the following conditions hold: r (e(t)).
Then, all signals of the closed-loop system are bounded, and the system output y tracks the reference signal y d with the desired preset performance under the following switching strategy.
Proof For t ∈ l q , t k+1 , the expectation of V (Z(t)) is obtained by combining (25). www.nature.com/scientificreports/ In terms of (H2) in Theorem 1, integrating it over [t k , t k+1 , by the Itô formula, one gets The number of stochastic switching for the time period [t k , t k+1 is denoted as N k . Since l 0 = t k , it follows from (27) and N(t, 0) represents the total number of switching in the time period [0, t) which is finite in switching system. The Lyapunov function V [j] i (Z) of each mode in subsystems satisfies (H1). So the following inequality (31) can be obtained.
One knows that N(t, 0) are bounded in switching system. Furthermore, since c, d, ϑ, η are selected positive numbers. Thus, following the similar discussion 11 , all the signals of the closed-loop system are bounded.

Numerical simulation
In this section, an example is given to illustrate the effectiveness of the proposed approach. Consider a dual switching time-delay system with two subsystems, each with two modes, i.e., M = 2, N = 2.
The two modes of the first determinate subsystem are as follows: # mode1: The two modes of the second determinate subsystem are as follows: # mode 1: # mode 2: We choose the reference signal y d = e −t − cos (t) , the time-delay τ 1 = 0.2s and τ 2 = 0.1s , β 1,1 (x 1 (t − τ 1 )) = x 2 1 (t − τ 1 ) + 1 and β 2,1 (x 1 (t − τ 1 )) + β 2,2 (x 2 (t − τ 2 )) = x 2 1 (t − τ 1 ) + x 2 2 (t − τ 1 ) . Define 1 = 2 = 1 and µ = 1 for all the subsystems and their modes. The parameters of prescribed performance control are selected as follows:  www.nature.com/scientificreports/ Figure 3 shows the switching signal γ (t) and control signal u(t) designed in Theorem 1. Figure 4 shows the adaptive law θ γ k i,σ of each determined subsystem. The letf subfigure of Fig. 5 shows the tracking effect of the system state output y on the specified reference signal y d . It can be seen that the system output y can track y d effectively under the control of the switching signal. The right subfigure of Fig. 5 shows the tracking error. The initial value of the tracking error s 1 (0) = 0.125 < ρ(0) = 1.8 . It is always within the performance bound (−δρ(t),δρ(t)) and tends to be zero, which guarantees the dynamic and steady-state performance of the controlled system. Control accuracy comparison without PPC and with PPC is also shown in Fig. 5. Under the control of the switching signal without PPC, although the tracking error gradually becomes smaller, it oscillates near the origin. The control accuracy is not as good as PPC, and steady-state performance cannot be obtained.

Conclusion
This paper discusses the adaptive neural network PPC problem for a class of dual switching nonlinear systems with time-delay. An adaptive NN controller based on a preset switching signal is proposed using the backstepping method, preset performance theory, and NN. The proposed scheme with preset performance can prove that all signals in the closed-loop system are bounded and the tracking error covers a small neighborhood of the origin. Simulation results verify the effectiveness of the proposed method. Our future work will focus on the PPC problem for double-switched stochastic nonlinear systems with unknown hysteresis.